In a large skillet over medium heat, brown beef with onion, green pepper, 1/2 teaspoon salt (I use less salt in this recipe) and pepper.
Remove meat mixture from skillet and drain fat. Reduce heat to low.
Melt butter; spread potatoes over butter and sprinkle with remaining salt. Top with beef mixture.
Combine eggs and milk; pour over all.
Cook covered until potatoes are tender, about 30 minutes.
Top with cheese; cover and heat until cheese is melted, about 5 minutes.
Cut into wedges or squares to serve.